Well-posed constrained evolution of 3+1 formulations of general relativity

Vasileios Paschalidis, Alexei Khokhlov, Igor Novikov

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5 Scopus citations


We present an analysis of well-posedness of constrained evolution of 3+1 formulations of general relativity. In this analysis we explicitly take into account the energy and momentum constraints as well as possible algebraic constraints on the evolution of high-frequency perturbations of solutions of Einstein's equations. In this respect, our approach is principally different from standard analyses of well-posedness of free evolution in general relativity. Our study reveals the existence of subsets of the linearized Einstein's equations that control the well-posedness of constrained evolution. It is demonstrated that the well-posedness of Arnowitt-Deser-Misner (ADM), Baumgarte-Shapiro-Shibata-Nakamura and other 3+1 formulations derived from the ADM formulation by adding combinations of constraints to the right-hand side of the ADM formulation and/or by linear transformation of the dynamical ADM variables depends entirely on the properties of the gauge. For certain classes of gauges we formulate conditions for well-posedness of constrained evolution. This provides a new basis for constructing stable numerical integration schemes for a classical ADM and many other 3+1 formulations of general relativity.

Original languageEnglish (US)
Article number024026
JournalPhysical Review D - Particles, Fields, Gravitation and Cosmology
Issue number2
StatePublished - 2007
Externally publishedYes

ASJC Scopus subject areas

  • Nuclear and High Energy Physics
  • Physics and Astronomy (miscellaneous)


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